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<!DOCTYPE proofs SYSTEM "proof3.dtd">

<?xml-stylesheet type="text/xsl" href="proof3.xsl"?>

<proofs number_of_proofs="1">

<source>AD-cs.out</source>

<heading><![CDATA[
Prover9 (32) version August-2006A, August 2006.
Process 11347 was started by mccune on cleo.thornwood,
Sat Aug 12 20:59:46 2006
The command was "/home/mccune/bin/prover9 -f cs.in AD.in".
]]></heading>

<proof number="1" length="24" max_count="25">

<comments><![CDATA[
% Proof 1 at 0.04 (+ 0.00) seconds: D.
% Length of proof is 24.
% Level of proof is 7.
% Maximum clause weight is 25.
% Given clauses 52.
]]></comments>

  <clause id="1">
    <literal><![CDATA[
      x * y != x * z
    ]]></literal>
    <literal><![CDATA[
      y = z
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="2">
    <literal><![CDATA[
      x * y != z * y
    ]]></literal>
    <literal><![CDATA[
      x = z
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="3">
    <literal><![CDATA[
      (x * y) * z = x * y * z
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="4">
    <literal><![CDATA[
      x * y * (y @ x) = y * x
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="5">
    <literal><![CDATA[
      (x @ y) @ z = u @ (v @ w)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="6">
    <literal><![CDATA[
      (A @ C) * (B @ C) != A * B @ C
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="7">
    <literal><![CDATA[
      A * B @ C != (A @ C) * (B @ C)
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[copy(6),flip(a)].">
      <j1 rule="copy" parents="6"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="8">
    <literal><![CDATA[
      x * y * z != x * y * u
    ]]></literal>
    <literal><![CDATA[
      z = u
    ]]></literal>
    <justification jstring="[para(3(a,1),1(a,1)),rewrite(3(4))].">
      <j1 rule="para" parents="3 1"/>
      <j2 rule="rewrite" parents="3"/>
    </justification>
  </clause>

  <clause id="10">
    <literal><![CDATA[
      x * (x @ x) = x
    ]]></literal>
    <justification jstring="[hyper(1,a,4,a)].">
      <j1 rule="hyper" parents="1 4"/>
    </justification>
  </clause>

  <clause id="11">
    <literal><![CDATA[
      x * y != z * x
    ]]></literal>
    <literal><![CDATA[
      z * (z @ x) = y
    ]]></literal>
    <justification jstring="[para(4(a,1),1(a,1)),flip(a)].">
      <j1 rule="para" parents="4 1"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="14">
    <literal><![CDATA[
      x * y * (y @ x) * z = y * x * z
    ]]></literal>
    <justification jstring="[para(4(a,1),3(a,1,1)),rewrite(3(2),3(5)),flip(a)].">
      <j1 rule="para" parents="4 3"/>
      <j2 rule="rewrite" parents="3 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      (x @ y) * z = z * (x @ y) * (u @ (v @ w))
    ]]></literal>
    <justification jstring="[para(5(a,1),4(a,1,2,2)),flip(a)].">
      <j1 rule="para" parents="5 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      (A * B @ C) * x != (A @ C) * (B @ C) * x
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[ur(2,b,7,a),rewrite(3(14))].">
      <j1 rule="ur" parents="2 7"/>
      <j2 rule="rewrite" parents="3"/>
    </justification>
  </clause>

  <clause id="24">
    <literal><![CDATA[
      (A @ C) * (B @ C) * x * (x @ (A * B @ C)) != x * (A * B @ C)
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[para(4(a,1),21(a,1)),flip(a)].">
      <j1 rule="para" parents="4 21"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="31">
    <literal><![CDATA[
      x * (x @ x) * y = x * y
    ]]></literal>
    <justification jstring="[para(10(a,1),3(a,1,1)),flip(a)].">
      <j1 rule="para" parents="10 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="33">
    <literal><![CDATA[
      (x @ y) * (z @ (u @ v)) = x @ y
    ]]></literal>
    <justification jstring="[para(5(a,1),10(a,1,2))].">
      <j1 rule="para" parents="5 10"/>
    </justification>
  </clause>

  <clause id="36">
    <literal><![CDATA[
      (x @ y) * z = z * (x @ y)
    ]]></literal>
    <justification jstring="[back_rewrite(17),rewrite(33(6))].">
      <j1 rule="back_rewrite" parents="17"/>
      <j2 rule="rewrite" parents="33"/>
    </justification>
  </clause>

  <clause id="90">
    <literal><![CDATA[
      (x @ x) * y = y
    ]]></literal>
    <justification jstring="[hyper(8,a,31,a)].">
      <j1 rule="hyper" parents="8 31"/>
    </justification>
  </clause>

  <clause id="109">
    <literal><![CDATA[
      x * ((y @ y) @ x) = x
    ]]></literal>
    <justification jstring="[para(90(a,1),4(a,1,2)),rewrite(90(5))].">
      <j1 rule="para" parents="90 4"/>
      <j2 rule="rewrite" parents="90"/>
    </justification>
  </clause>

  <clause id="111">
    <literal><![CDATA[
      ((x @ y) @ z) * u = u
    ]]></literal>
    <justification jstring="[para(5(a,2),90(a,1,1))].">
      <j1 rule="para" parents="5 90"/>
    </justification>
  </clause>

  <clause id="150">
    <literal><![CDATA[
      x * (y @ x) * y * z = y * x * z
    ]]></literal>
    <justification jstring="[para(14(a,1),14(a,1,2)),rewrite(111(7))].">
      <j1 rule="para" parents="14 14"/>
      <j2 rule="rewrite" parents="111"/>
    </justification>
  </clause>

  <clause id="160">
    <literal><![CDATA[
      x * (y @ (z @ u)) = x
    ]]></literal>
    <justification jstring="[para(5(a,1),109(a,1,2))].">
      <j1 rule="para" parents="5 109"/>
    </justification>
  </clause>

  <clause id="175">
    <literal><![CDATA[
      (A @ C) * (B @ C) * x != x * (A * B @ C)
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[back_rewrite(24),rewrite(160(13))].">
      <j1 rule="back_rewrite" parents="24"/>
      <j2 rule="rewrite" parents="160"/>
    </justification>
  </clause>

  <clause id="280">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[ur(11,b,175,a(flip)),rewrite(36(11),3(11),150(13),4(8),3(10)),xx(a)].">
      <j1 rule="ur" parents="11 175"/>
      <j2 rule="rewrite" parents="36 3 150 4 3"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
